Volume 13, Issue 6
An $h$-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers

Huanying Bian, Yedan Shen & Guanghui Hu

Adv. Appl. Math. Mech., 13 (2021), pp. 1418-1440.

Published online: 2021-08

Preview Full PDF 26 5430
Export citation
  • Abstract

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media. In this paper, a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon. Towards the efficiency and accuracy of the numerical simulations, a predictor-corrector process is used for resolving the nonlinearity of the equation, and an $h$-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region, in which a heuristic a posteriori error indicator is designed for the purpose. In numerical simulations, a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method. The effectiveness of the $h$-adaptive method is also successfully demonstrated by numerical experiments. Finally the mechanism on generating fingers is discussed by numerically studying several examples.


  • Keywords

Non-equilibrium Richard equation, $h$-adaptive mesh method, a posteriori error estimation, fingering phenomenon, porous media flow.

  • AMS Subject Headings

65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{AAMM-13-1418, author = {Bian , Huanying and Shen , Yedan and Hu , Guanghui}, title = {An $h$-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {6}, pages = {1418--1440}, abstract = {

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media. In this paper, a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon. Towards the efficiency and accuracy of the numerical simulations, a predictor-corrector process is used for resolving the nonlinearity of the equation, and an $h$-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region, in which a heuristic a posteriori error indicator is designed for the purpose. In numerical simulations, a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method. The effectiveness of the $h$-adaptive method is also successfully demonstrated by numerical experiments. Finally the mechanism on generating fingers is discussed by numerically studying several examples.


}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0218}, url = {http://global-sci.org/intro/article_detail/aamm/19429.html} }
TY - JOUR T1 - An $h$-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers AU - Bian , Huanying AU - Shen , Yedan AU - Hu , Guanghui JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1418 EP - 1440 PY - 2021 DA - 2021/08 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0218 UR - https://global-sci.org/intro/article_detail/aamm/19429.html KW - Non-equilibrium Richard equation, $h$-adaptive mesh method, a posteriori error estimation, fingering phenomenon, porous media flow. AB -

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media. In this paper, a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon. Towards the efficiency and accuracy of the numerical simulations, a predictor-corrector process is used for resolving the nonlinearity of the equation, and an $h$-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region, in which a heuristic a posteriori error indicator is designed for the purpose. In numerical simulations, a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method. The effectiveness of the $h$-adaptive method is also successfully demonstrated by numerical experiments. Finally the mechanism on generating fingers is discussed by numerically studying several examples.


Huanying Bian, Yedan Shen & GuanghuiHu. (1970). An $h$-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers. Advances in Applied Mathematics and Mechanics. 13 (6). 1418-1440. doi:10.4208/aamm.OA-2020-0218
Copy to clipboard
The citation has been copied to your clipboard